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Properties

Label 2-462-77.62-c1-0-9
Degree $2$
Conductor $462$
Sign $0.875 + 0.483i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 − 0.951i)4-s + (1.12 + 1.54i)5-s + (0.809 − 0.587i)6-s + (0.488 − 2.60i)7-s + (0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s − 1.90·10-s + (−0.215 − 3.30i)11-s + i·12-s + (−2.19 − 1.59i)13-s + (1.81 + 1.92i)14-s + (−0.589 − 1.81i)15-s + (−0.809 + 0.587i)16-s + (−3.34 + 2.42i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−0.549 − 0.178i)3-s + (−0.154 − 0.475i)4-s + (0.501 + 0.690i)5-s + (0.330 − 0.239i)6-s + (0.184 − 0.982i)7-s + (0.336 + 0.109i)8-s + (0.269 + 0.195i)9-s − 0.603·10-s + (−0.0648 − 0.997i)11-s + 0.288i·12-s + (−0.610 − 0.443i)13-s + (0.485 + 0.514i)14-s + (−0.152 − 0.468i)15-s + (−0.202 + 0.146i)16-s + (−0.810 + 0.588i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.875 + 0.483i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.875 + 0.483i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.910967 - 0.234714i\)
\(L(\frac12)\) \(\approx\) \(0.910967 - 0.234714i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 + (0.951 + 0.309i)T \)
7 \( 1 + (-0.488 + 2.60i)T \)
11 \( 1 + (0.215 + 3.30i)T \)
good5 \( 1 + (-1.12 - 1.54i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (2.19 + 1.59i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.34 - 2.42i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.23 + 3.79i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6.88T + 23T^{2} \)
29 \( 1 + (-6.78 + 2.20i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.61 + 2.22i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.321 - 0.988i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.51 + 7.73i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.21iT - 43T^{2} \)
47 \( 1 + (-7.21 - 2.34i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.96 + 2.88i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.13 + 1.01i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.21 - 4.51i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 1.75T + 67T^{2} \)
71 \( 1 + (10.1 - 7.36i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.13 - 6.57i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.00 - 1.38i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.15 + 0.839i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 7.07iT - 89T^{2} \)
97 \( 1 + (-3.78 + 5.21i)T + (-29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.69204021615496527534524005284, −10.36245217068949700699470882775, −9.160836524434348475868567485831, −8.119674646783600380771560191369, −7.08079541681163516414792580680, −6.54895668259723213930631653868, −5.51220839972794544263142083049, −4.40632784145972066889655877920, −2.72324036696775054205408251533, −0.77742209278590906539886295098, 1.50362825771223867096535687972, 2.75403455552666468553926288580, 4.63120243081217447002495279667, 5.11651037272219408252173780601, 6.44899850873811470792628225355, 7.54512122609533921251564773936, 8.862092684832736054226887649905, 9.317331541955953196912061277712, 10.14350413378089842491657162070, 11.16003245735554152252840903554

Graph of the $Z$-function along the critical line